Pappus of Alexandria ( ; Ancient Greek: Πάππος ὁ Ἀλεξανδρεύς; c. 290 – c. 350 AD) was a Greek mathematician of late antiquity known for his Synagoge (Συναγωγή) or Collection (c. 340), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known about his life except for what can be found in his own writings, many of which are lost. Pappus apparently lived in Alexandria, where he worked as a mathematics teacher to higher level students, one of whom was named Hermodorus.
The Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics that were part of the ancient mathematics curriculum, including geometry, astronomy, and mechanics.
Pappus was active in a period generally considered one of stagnation in mathematical studies, where, to some, he stands out as a remarkable exception and, to others, as an exemplar of ills that halted the progress of Greek science. In many respects, his fate strikingly resembles that of Diophantus', originally of limited importance but becoming very influential in the late Renaissance and Early Modern periods.
In his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time (but see below) when he himself wrote. If no other date information were available, all that could be known would be that he was later than Ptolemy (died c. 168 AD), whom he quotes, and earlier than Proclus (born c. 411), who quotes him.
The 10th century Suda states that Pappus was of the same age as Theon of Alexandria, who was active in the reign of Emperor Theodosius I (372–395). A different date is given by a marginal note to a late 10th-century manuscript (a copy of a chronological table by the same Theon), which states, next to an entry on Emperor Diocletian (reigned 284–305), that "at that time wrote Pappus".
However, a verifiable date comes from the dating of a solar eclipse mentioned by Pappus himself. In his commentary on the Almagest he calculates "the place and time of conjunction which gave rise to the eclipse in Tybi in 1068 after Nabonassar". This works out as 18 October 320, and so Pappus must have been active around 320.
The great work of Pappus, in eight books and titled Synagoge or Collection, has not survived in complete form: the first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus: Χωρογραφία οἰκουμενική (Chorographia oikoumenike or Description of the Inhabited World), a commentary on the thirteen books of Ptolemy's Almagest (of which the part on books 5 and 6 survives), Ποταμοὺς τοὺς ἐν Λιβύῃ (The Rivers in Libya), and Ὀνειροκριτικά (The Interpretation of Dreams). Pappus himself mentions another commentary of his own on the Ἀνάλημμα (Analemma) of Diodorus of Alexandria. Pappus also wrote commentaries on Euclid's Elements (of which fragments are preserved in Proclus and the Scholia, while that on the tenth Book has been found in an Arabic manuscript), and on Ptolemy's Ἁρμονικά (Harmonika).
Federico Commandino translated the Collection of Pappus into Latin in 1588. The German classicist and mathematical historian Friedrich Hultsch (1833–1908) published a definitive three-volume presentation of Commandino's translation with both the Greek and Latin versions (Berlin, 1875–1878). Using Hultsch's work, the Belgian mathematical historian Paul ver Eecke was the first to publish a translation of the Collection into a modern European language; his two-volume, French translation has the title Pappus d'Alexandrie. La Collection Mathématique. (Paris and Bruges, 1933).
Pappus's Collection contains an account, systematically arranged, of the most important results obtained by his predecessors and notes explanatory of, or extending, previous discoveries. These discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the various books valuable, for they set forth clearly an outline of the contents and the general scope of the subjects to be treated. In these introductions, the style of Pappus's writing is excellent and even elegant the moment he is free from the shackles of mathematical formulae and expressions. Heath also found his characteristic exactness made his Collection "a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us".
The surviving portions of Collection can be summarized as follows.
Book I has been lost. Book I, like Book II, may have been concerned with arithmetic, as Book III is clearly introduced as beginning a new subject.
The whole of Book II (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition) discusses a method of multiplication from an unnamed book by Apollonius of Perga. The final propositions deal with multiplying together the numerical values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2×1054 and 2×1038.
Book III contains geometrical problems, plane and solid. It may be divided into five sections:
On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the cube, reduced by Hippocrates of Chios to the former. Pappus gives several solutions of this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one.
On the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure. This serves as an introduction to a general theory of means, of which Pappus distinguishes ten kinds, and gives a table representing examples of each in whole numbers.
On a curious problem suggested by Euclid I. 21.
On the inscribing of each of the five regular polyhedra in a sphere. Here Pappus observed that a regular dodecahedron and a regular icosahedron could be inscribed in the same sphere such that their vertices all lay on the same 4 circles of latitude, with 3 of the icosahedron's 12 vertices on each circle, and 5 of the dodecahedron's 20 vertices on each circle. This observation has been generalised to higher dimensional dual polytopes.
An addition by a later writer on another solution of the first problem of the book.